Description
This course is a seminar in topology. The main mathematical goal is to learn about the fundamental group, homology and cohomology. The main non-mathematical goal is to obtain experience giving math talks. Lectures will be delivered by the students, with two students speaking at each class. There are no exams. There will be some homework assignments and a final paper.
Seminar leader
Andrew Snowden
e-mail: asnowden at math dot mit dot edu
Office: 2-175
Office hours by appointment
Time and location
The seminar typically meets Monday, Wednesday and Friday from 12pm to 1pm in room 2-139. See the calendar for exceptions. Practice lectures will also take place in room 2-139.
Textbooks
We will mainly use Hatcher's “Algebraic topology.” This book is available for free online at Hatcher's webpage. (It is also available in print.) We may also make some use of Massey's “A basic course in algebraic topology,” which is published by Springer in the Graduate Texts in Math series (GTM 127).
Grading
The final grade is determined as follows:
- 60% — Lectures and participation
- 30% — Final paper
- 10% — Problem sets
Attendance is mandatory. Every three missed classes will result in the drop of a letter grade; thus one can miss up to two classes with no effect on the grade. The classes on 2/2 and 2/4 will not count towards this. Classes missed for a valid medical excuse will also not count towards this. If you know you will miss a class for some reason, e-mail me a day or two in advance and we can try to work something out.
Lectures and participation
Each class two students will give lectures. Each lecture should be about 25 minutes long. Individual lectures will not be graded, but lectures make up a good portion of the final grade. In evaluating your lectures, I will look at their clarity, organization and preparedness. I will also consider how your lectures improve over the course of the semester.
You will give a practice lecture to a small audience (consisting of me, Susan Ruff and the other student lecturing in the same class as you) before your first lecture.
Each lecturer will give one or two exercises relevant to the material being presented. These exercises, and their solutions, should be e-mailed to me as a Latex file. The exercises can be stated during lecture, though this is not necessary. It's ok if the exercises come from a book (although it'd be preferable if they did not, or at least if they were slightly modified), but be sure to give proper attribution.
As a member of the audience, I'd like you to write a few comments on each lecture you observe. I'm not asking for any kind of lengthy analysis; it would be enough to point out that the lecturer is writing too small. However, make sure the comments are useful — don't just say “that proof was good,” say why. I will collect these comments at the end of class and e-mail them to the lecturer so that they can have some feedback. (The lecturer will not know who made which comments.)
Homework
There will be approximately four problem sets. These will count towards the final grade. Solutions are to be written in Latex. You may work together on the problem sets, but everyone must write up their own solutions.
There will also be exercise sets, mainly composed of exercises given by lecturers. These are optional and do not have to be turned in. If you are intested in learning the material, it is probably a good idea to do at least some of the exercises.
Final paper
The final paper is an exposition of a topic in algebraic topology that we will not cover in the seminar. It must be at least 10 pages long and written in Latex. Topics will be selected for the papers in March. A first draft is due in April, and a final draft two weeks later. In the final six or so meetings of class, students will give talks on their final papers.
February | ||||
W | Feb 2 | First meeting of class. | ||
M | Feb 21 | President's Day, no class. But there is class tomorrow. | ||
T | Feb 22 | Class meets today. | ||
Problem set 1 due. | ||||
March | ||||
M | Mar 7 | Deadline for selecting topic for final paper. | ||
F | Mar 11 | Andrew is gone, class meets as usual. | ||
M | Mar 14 | Andrew is gone, class meets as usual. | ||
W | Mar 16 | Andrew is gone, class meets as usual. | ||
F | Mar 18 | Andrew is gone, class meets as usual. | ||
Problem set 2 due. | ||||
M | Mar 21 | Spring break. | ||
W | Mar 23 | Spring break. | ||
F | Mar 25 | Spring break. | ||
April | ||||
M | Apr 11 | First draft of paper due. | ||
M | Apr 18 | Patriot's Day, no class. | ||
M | Apr 25 | Final draft of paper due. | ||
May | ||||
W | May 11 | Final meeting of class. |
► February | ||||
W | Feb 2 | Andrew | Organizational meeting | |
F | Feb 4 | Andrew | Introduction to the fundamental group | |
M | Feb 7 | Scott | Paths and homotopies | |
Umut | The fundamental group | |||
W | Feb 9 | Kyle | The fundamental group of the circle | |
JJ | Applications of previous lecture | |||
F | Feb 11 | Marcel | Contractible and simply connected spaces | |
Danny | The fundamental group of a product | |||
M | Feb 14 | John | Functoriality of the fundamental group | |
Noah | Homotopy equivalences | |||
W | Feb 16 | Rafael | The fundamental group of S^{1} ∨ S^{1} | |
Gabriel | Amalgamated free products | |||
F | Feb 18 | Aldo | van Kampen's theorem | |
Andrew | van Kampen's theorem (continued) | |||
M | Feb 21 | President's Day, no class. But there is class tomorrow. | ||
T | Feb 22 | Andrew | Introduction to covering spaces | |
W | Feb 23 | Rafael | The universal cover | |
JJ | The universal cover (continued) | |||
F | Feb 25 | Noah | Lifting properties | |
Gabriel | Lifting properties (continued) | |||
M | Feb 28 | Danny | Existence of covers | |
Umut | The Galois correspondence |
References: Hatcher §1.1, Massey §II.2.
References: Hatcher §1.1, Massey §II.3.
References: Hatcher §1.1, Massey §II.5.
References: Hatcher §1.1, Massey §II.6.
References: ?
References: Hatcher §1.1, Massey §II.7.
References: Hatcher §1.1, Massey §II.4.
References: Hatcher §1.1, Massey §II.8.
References: Hatcher pp. 63–65.
References: Hatcher pp. 63–65.
References: Hatcher Prop. 1.30 and Prop. 1.31.
References: Hatcher Prop. 1.33 and Prop. 1.34.
References: Hatcher Prop. 1.36 and Prop. 1.39.
References: Hatcher Prop. 1.37 and Thm. 1.38.
◄ ► March | ||||
W | Mar 2 | Kyle | Category theory | |
Marcel | The Galois correspondence in categorical form | |||
F | Mar 4 | John | Covering spaces of S^{1} ∨ S^{1} | |
Aldo | Quotients by finite groups | |||
M | Mar 7 | Andrew | Overview of homology | |
W | Mar 9 | Scott | Chains and the boundary operator | |
Noah | Definition of homology and first calculations | |||
F | Mar 11 | John | Chain complexes | |
Rafael | Functoriality of homology | |||
M | Mar 14 | Aldo | The long exact sequence | |
Danny | Relative homology | |||
W | Mar 16 | Scott | Excision | |
JJ | Homology of a quotient | |||
F | Mar 18 | Umut | Proof of Prop. 2.21, part 1 | |
Gabriel | Proof of Prop. 2.21, part 2 | |||
M | Mar 21 | Spring break, no class. | ||
W | Mar 23 | Spring break, no class. | ||
F | Mar 25 | Spring break, no class. | ||
M | Mar 28 | Andrew | Review | |
Andrew | Naturality of connecting homomorphisms | |||
W | Mar 30 | Kyle | Axioms for homology | |
Rafael | The Mayer–Vietoris sequence |
References: Hatcher pp. 68–70.
References: Hatcher p. 57.
References: Hatcher Prop. 1.40.
References: Hatcher p. 103 and p. 108.
References: Hatcher p. 108, Prop. 2.7 and Prop. 2.8.
References: Some discussion in Hatcher pp. 110–113.
References: Hatcher pp. 110–113, esp. Thm. 2.10.
References: Some discussion in Hatcher pp. 115–118.
References: Hatcher pp. 115–118, esp. Example 2.17.
References: Hatcher p. 119 (esp. Thm. 2.20) and p. 124.
References: Hatcher p. 110, Cor. 2.14 and Prop. 2.22.
References: Hatcher pp. 119–124.
References: Hatcher pp. 119–124.
References: Hatcher pp. 127–128.
References: Hatcher pp. 160–162.
References: Hatcher pp. 153–154 and pp. 161–162.
◄ ► April | ||||
F | Apr 1 | Andrew | Homology with coefficients | |
Danny | The universal coefficient theorem | |||
M | Apr 4 | Noah | CW complexes, part 1 | |
Umut | CW complexes, part 2 | |||
W | Apr 6 | JJ | CW homology, part 1 | |
Scott | CW homology, part 2 | |||
F | Apr 8 | Aldo | Definition of cohomology | |
Kyle | Overview of formal properties | |||
M | Apr 11 | John | The cup product, part 1 | |
Gabriel | The cup product, part 2 | |||
W | Apr 13 | Umut | The Kunneth formula, part 1 | |
Andrew | The Kunneth formula, part 2 | |||
F | Apr 15 | Andrew | Overview of Poincare duality | |
Kyle | Orientations | |||
M | Apr 18 | Patriot's Day, no class. | ||
W | Apr 20 | JJ | The fundamental class, part 1 | |
Rafael | The fundamental class, part 2 | |||
F | Apr 22 | Andrew | Direct limits | |
Scott | Cohomology with compact support | |||
M | Apr 25 | Aldo | The cap product | |
Danny | Statement of Poincare duality | |||
W | Apr 27 | John | Proof of Lemma 3.36 | |
Noah | Proof of Poincare duality | |||
F | Apr 29 | Andrew | Closing lecture |
References: Hatcher pp. 153–154.
References: Hatcher Appendix 3.A.
References: Hatcher, Appendix.
References: Hatcher, Appendix.
References: Hatcher pp. 137–141.
References: Hatcher pp. 141–146.
References: Hatcher pp. 197–198.
References: Hatcher pp. 199–204.
References: Hatcher pp. 206–207, Prop. 3.10, Thm. 3.14.
References: Hatcher Example 3.13.
References: Hatcher pp. 218–219, Appendix 3.B.
References: Hatcher pp. 220–221.
References: Hatcher p. 231.
References: Hatcher pp. 233–235.
References: Hatcher p. 236.
References: Hatcher p. 327.
References: Hatcher pp. 243–244.
References: Hatcher pp. 242–245.
References: Hatcher pp. 239–241.
References: Hatcher Thms 3.30 and 3.35.
References: Hatcher Lemma 3.36.
References: Hatcher pp. 247–248.
◄ May | ||||
M | May 2 | John | Sheaf cohomology | |
Noah | Morse theory | |||
W | May 4 | JJ | Hopf fibrations | |
Aldo | The Gauss—Bonnet theorem | |||
F | May 6 | Scott | K-theory | |
Umut | De Rham cohomology | |||
M | May 9 | Danny | The Hurewicz isomorphism | |
Rafael | Orbifold fundamental groups | |||
W | May 11 | Kyle | Simplicial sets |
Instructions for problem sets
- You may work together, but everybody must write-up their own solutions.
- Solutions may use results from class, but should otherwise be self-contained. You may freely use basic results without proof or reference.
- Feel free to consult books to review material needed to complete the problems, but don't look up solutions to the problems!
- Solutions must be written in Latex, printed, stapled and handed-in at the beginning of class. Please make your solutions readable; feel free to state intermediate results as lemmas. It may be cleanest to make each problem its own section. Also, set the margins to something reasonable (the default margins in Latex are not reasonable).
- Each problem will be graded out of 10 points, unless otherwise indicated.
- E-mail me if a problem statement is confusing or incomplete.
Problem sets
- Problem Set 3, due Monday, May 9th.
- Problem Set 2, due Friday, March 18th. Solutions (partial).
- Problem Set 1, due Tuesday, February 22nd. Solutions.
Lecture notes
For one round of lectures, the students submitted typed up lecture notes. We decided not to continue this, as it took too much time and books already have the same material.
Exercises
For a while, students composed exercises related to the material on which they were lecturing. These exercises were intended as practice for the other students with the material, and were not required to be completed.
Concering the final paper
This document gives a list of mistakes that were common in the drafts of your final papers. Every draft that I read contained at least one of these mistakes, so I recommend that you all look through this and apply what it says to your writing.
A sample file
This note (and its tex source) gives some examples of how to do things like matrices and commutative diagrams, and also discusses some things that you should and should not do in latex (repeated in abridged form below).
Last updated: 3/3/11.
Stuff not to do
These remarks are elaborated upon in the above file, but put here for convenience and emphasis:
- Do not begin sentences (or phrases) with math. Do not put math next to math.
- Multi-letter operators and functions, like sin, should not be italicized.
- Use the correct size parthenses.
- When multiple formulas are put into a single displaymath, put space between them.
- The default margins are much too large; adjust them.
The final paper is an approximately 10 page exposition on a topic in algebraic topology not covered in our seminar. The paper must be written in Latex (or some other flavor of Tex).
You must select the topic for your paper by March 7th. I'd prefer that no two of you do the same topic, so if there's something you'd really like to do you should tell me soon. When you know what you want to do, just send me an e-mail.
Below is a list of possible ideas for topics. You're free to choose something not on this list, but run it by me first.
Concrete topics
- Hopf fibrations. The Hopf fibrations are three specific maps of spheres defined using projective spaces over the complex numbers, quaternions and octonions. They were the first non-trivial elements of higher homotopy groups of spheres to be discovered. This topic has been taken.
- Lens spaces. The lens spaces are an explicit family of 3-dimensional manifolds indexed by pairs of integers. They provided the first example of a pair of compact manifolds which are homotopy equivalent but not homeomorphic.
- PSL_{2}(Z) as a free product. Using some algebraic topology, one can give a nice proof that PSL_{2}(Z) (roughly the group 2×2 integer matrices) is the free product of a cyclic group of order 2 and a cyclic group of order 3. This topic has been taken.
Homology theories
- Borel–Moore homology and cohomology with supports. These variants of homology and cohomology are useful when dealing with non-compact spaces. They allow one to extend Poincaré duality to the setting of non-compact manifolds.
- Simplicial and cellular homology. Simplicial complexes give a very elementary and combinatorial way to describe topological spaces. Simplicial homology describes how to compute the homology of a space that is described as a simplicial complex. CW complexes also provide a way to describe topological spaces, one that is less elementary but more flexible than that provided by simplicial complexes. Cellular homology describes how to compute the homology of space described as a CW complex. This topic has been removed.
- de Rham cohomology. This is a cohomology theory for smooth manifolds defined in terms of differential forms. The main theorem, the de Rham isomorphism theorem, states that de Rham cohomology is isomorphic to singular cohomology (with real coefficients). This is remarkable since the definition uses the smooth structure on the manifold but the final product is only dependent on the topology. p-adic analogues of this result have been very important in number theory in recent years. This topic has been taken.
- Sheaf theory. Sheaves are to topological spaces what modules are to rings. There is a notion of cohomology for a sheaf on a topological space; the cohomology of the so-called “constant sheaf” gives the singular cohomology of the space. Sheaf theory is indispensable in modern algebraic geometry. This topic has been taken.
- K-theory. Given a space X, a group K_{0}(X) is constructed using complex vector bundles on X. Groups K_{n}(X) can be constructed in various manners. The K-groups are not isomorphic to the singular cohomology groups. K-theory forms what is called an “extraordinary cohomology theory.&rdquo The most important result is Bott periodicity, which states that the K-groups are periodic with period 2. This topic has been taken.
Categorical methods
- Simplicial sets. The theory of simplicial sets offers a model of homotopy theory without using topological spaces. Instead, it relies on certain diagrams of sets. Homology can be described elegantly in this theory; in fact, it essentially amounts to taking the free abelian group on the simplicial set. This topic has been taken.
- The fundamental groupoid. The fundamental group of a topological space depends on the choice of a basepoint, which can be inconvenient. The fundamental groupoid is something like the fundamental group, but does not depend on any choice. However, it is a category instead of a group. This can be extended to higher homotopy groups through the use of higher categories.
- The derived category. The Kunneth formula states that H_{•}(X × Y)=H_{•}(X) ⊗ H_{•}(Y), up to some error terms involving Tor's. Similarly, the universal coefficient theorem states that H_{•}(X; R)=H_{•}(X; Z) ⊗ R, again up to some error terms involving Tor's. Derived categories provide a way to eliminate these error terms, which is very desirable. The basic idea of derived categories is to systematically work with the complexes that compute homology, rather than homology itself. Derived categories are very important in a wide variety of subjects these days.
Classifying spaces
- Eilenberg–MacLane spaces. Let G be a group and let n≥1 be an integer; if n>1 then assume that G is commutative. One can then show that there is a unique homotopy type X with π_{n}(X)=G and π_{k}(X)=1 for k≠n. This homotopy type is usually denoted K(G, n) and called an Eilenberg–MacLane space. One reason that these spaces are interesting is that they represent cohomology: giving an element of H^{n}(X; G) is the same as giving a homotopy class of maps X → K(G, n).
- Group cohomology. Let G be a group. As stated above, there is an associated homotopy type K(G, 1). The (co)homology of K(G, 1) is an invariant of the group G, called group (co)homology. One can give a purely algebraic construction of group (co)homology, however, the topological perspective is often useful as well.
- The classifying space for vector bundles. Let n≥1 be an integer. There is then a space B with the following property: homotopy classes of maps from an arbitrary space X into B are canonically in bijection with complex vector bundles of rank n on X. The space B is infinite dimensional, but can be described explicitly as a Grassmannian. Using classifying spaces, one can associate cohomology classes to vector bundles; this is the theory of characteristic classes.
Higher homotopy groups
- The long exact sequence. A fibration is the analogue in the world of homotopy theory to the concept of a short exact sequence. Given a fibration F → X → B, there is a long exact sequence relating the homotopy groups of F, X and B. This can be used to calculate some higher homotopy groups. This topic has been taken.
- The Freudenthal suspension theorem. This result states that, in certain cases, homotopy groups of suspensions of a space stabilize; precisely, π_{n+k}(S^{k}X) is independent of k when k is large. This leads to the concept stable homotopy groups, and a whole stable homotopy theory.
- The Hurewicz homomorphism. There is a natural map from homotopy to singular homology, called the Hurewicz homomorphism. Hurewicz showed that for a simply connected space, the first non-zero homotopy and homology groups are isomorphic via this map. This topic has been taken.
Other topics
- Spectral sequences. As mentioned above, associated to a fibration of spaces is a long exact sequence of homotopy groups. The behvior of homology in a fibration is not as simple — it is described by an algebraic construct called a spectral sequence. One can use spectral sequences to compute the homology of certain spaces, such as Lie groups and some loop spaces.
- Steenrod squares. Steenrod constructed a natural map H^{n}(X; F_{2}) → H^{n+i}(X F_{2}) for each i, which is now called the Steenrod square Sq^{i}. This gives the cohomology of a space with F_{2} coefficients extra structure, and allows one to distinguish spaces using cohomology that one could not before.
- Morse theory. This theory gives a way to decompose a manifold using critical points of functions on it. John Milnor has a nice book on the subject. This topic has been taken.
- The Gauss–Bonnet theorem. Given a surface in Euclidean space (or, more abstractly, a Riemannian manifold of dimension 2), one can speak of the curvature of the surface at a point. The Gauss–Bonnet theorem states that the total curvature (the integral over the surface of the curvature at each point) is equal to the Euler characteristic of the surface. Like the de Rham theorem mentioned above, this is remarkable since it shows that the total curvature of a surface is a topological invariant, even though this is not apparent in its definition. This topic has been taken.